Nonautonomous solitons in external potentials

Novel soliton solutions for the nonautonomous nonlinear Schr?dinger equation models with linear and harmonic oscillator potentials substantially extend the concept of classical solitons and generalize it to the plethora of nonautonomous solitons that interact elastically and generally move with varying amplitudes, speeds, and spectra adapted both to the external potentials and to the dispersion and nonlinearity variations. The nonautonomous soliton concept can be applied to different physical systems, from hydrodynamics and plasma physics to nonlinear optics and matter waves, and offer many opportunities for further scientific studies.     

Solitons of two-component Bose-Einstein condensates modulated in space and time

In this Letter we present soliton solutions of two coupled nonlinear Schrödinger equations modulated in space and time. The approach allows us to obtain solitons for a large variety of solutions depending on the nonlinearity and potential profiles. As examples we show three cases with soliton solutions: a solution for the case of a potential changing from repulsive to attractive behavior, and the other two solutions corresponding to localized and delocalized nonlinearity terms, respectively.     

An exact (2 + 1)-dimensional optical soliton with spatially modulated nonlinearity and an external potential

An exact (2 + 1)-dimensional spatial optical soliton of the nonlinear Schrödinger equation with a spatially modulated nonlinearity and a special external potential is discovered in an inhomogeneous nonlinear medium, by utilizing the similarity transformation. Exact analytical solutions are constructed by the products of Whittaker functions and the bright and dark soliton solutions of the standard stationary nonlinear Schrödinger equation. Some examples of such composed solutions are given, in which these spatial solitons display different localized structures. Numerical calculation shows that the soliton is stable in propagating over long distances, thus also confirming the validity of the exact solution.     

Analytical soliton solutions for the general nonlinear Schrödinger equation including linear and nonlinear gain (loss) with varying coefficients

An improved homogeneous balance principle and an F-expansion technique are used to construct analytical solutions to the generalized nonlinear Schrödinger equation with distributed coefficients and linear and nonlinear gain (or loss). For limiting parameters, these periodic wave solutions acquire the form of localized spatial solitons. Such solutions exist under certain conditions, and impose constraints on the functions describing dispersion, nonlinearity, and gain (or loss). We present a few characteristic examples of periodic wave and soliton solutions with physical relevance.     

Non-autonomous matter-wave solitons in hybrid atomic–molecular Bose–Einstein condensates with tunable interactions and harmonic potential

In this paper, we investigate matter-wave solitons in hybrid atomic–molecular Bose–Einstein condensates with tunable interactions and external potentials. Three types of time-modulated harmonic potentials are considered and, for each of them, two groups of exact non-autonomous matter-wave soliton solutions of the coupled Gross–Pitaevskii equation are presented. Novel nonlinear structures of these non-autonomous matter-wave solitons are analyzed by displaying their density distributions. It is shown that the time-modulated nonlinearities and external potentials can support exact non-autonomous atomic–molecular matter-wave solitons.     

Optical soliton perturbation for time fractional resonant nonlinear Schrödinger equation with competing weakly nonlocal and full nonlinearity

In this article, we retrieve optical soliton solutions of the perturbed time fractional resonant nonlinear Schrödinger equation having competing weakly nonlocal and full nonlinearity. We study the equation for two different forms of nonlinearity, namely Kerr law and anti-cubic law. The F-expansion method along with fractional complex transformation is used to obtain the optical solitons. Moreover, the existence of these solitons are guaranteed with the constraint relations between the model coefficients and the traveling wave frequency coefficient.     

Nonautonomous bright matter-wave solitons and soliton collisions in Fourier-synthesized optical lattices

We present analytical bright multisoliton solutions to the generalized nonautonomous nonlinear Schrödinger equation with time- and space-dependent distributed coefficients in Fourier-synthesized optical lattice potential based on the similarity transformation technique. Such solutions exist in certain constraint conditions on the coefficients depicting dispersion, nonlinearity, and gain (or loss). Various shapes of bright solitons and interesting interactions between two solitons are observed, including soliton trains, collapse and revival of condensates, and two periodic M-shape solitons with collision. Phenomena of a few solitons and physical applications of interest to the field are discussed.     

非均匀系统中非自治孤子的色散和非线性管理

基于包含自发拉曼散射和外电光调制效应的非自治非线性薛定谔方程,采用简单的变换方法,解析研究了三种非均匀系统中非自治孤子的管理和传输。结果发现,在非均匀的非线性渐增或色散渐减光纤系统中都存在精确的啁啾非自治孤子解,都可以实现孤子的放大和压缩,但具有不同的速度、频移和啁啾特性;而在非均匀的色散和非线性均渐减光纤系统中,可以支持无啁啾的非自治孤子,该孤子具有不变的脉宽和振幅以及振荡衰减的速度,孤子的频移仅由自发拉曼散射决定。同时,数值模拟结果进一步证实在三种非均匀管理系统中都可支持非自治孤子的传输。该研究结果为实际非均匀孤子管理系统中实现孤子的压缩和传输提供了理论依据。     

Soliton-like solutions for the nonlinear schr?dinger equation with variable quadratic hamiltonians

We construct one-soliton solutions for the nonlinear Schr¨odinger equation with variable quadratic Hamiltonians in a unified form by taking advantage of the complete (super) integrability of generalized harmonic oscillators. The soliton-wave evolution in external fields with variable quadratic potentials is totally determined by the linear problem, like motion of a classical particle with acceleration, and the (self-similar) soliton shape is due to a subtle balance between the linear Hamiltonian (dispersion and potential) and nonlinearity in the Schr¨odinger equation by the standards of soliton theory. Most linear (hypergeometric, Bessel) and a few nonlinear (Jacobian elliptic, second Painlev′e transcendental) classical special functions of mathematical physics are linked together through these solutions, thus providing a variety of nonlinear integrable cases. Examples include bright and dark solitons and Jacobi elliptic and second Painlev′e transcendental solutions for several variable Hamiltonians that are important for research in nonlinear optics, plasma physics, and Bose–Einstein condensation. The Feshbach-resonance matter-wave-soliton management is briefly discussed from this new perspective.     

Nonautonomous bright solitons and soliton collisions in a nonlinear medium with an external potential

We present exact bright multi-soliton solutions of a generalized nonautonomous nonlinear Schrdinger equation with time-and space-dependent distributed coefficients and an external potential which describes a pulse propagating in nonlinear media when its transverse and longitudinal directions are nonuniformly distributed.Such solutions exist in certain constraint conditions on the coefficients depicting dispersion,nonlinearity,and gain(loss).Various shapes of bright solitons and interesting interactions between two solitons are observed.Physical applications of interest to the field and stability of the solitons are discussed.     

Two-dimensional matter-wave solitons and vortices in competing cubic-quintic nonlinear lattices

The nonlinear lattice — a new and nonlinear class of periodic potentials — was recently introduced to generate various nonlinear localized modes. Several attempts failed to stabilize two-dimensional (2D) solitons against their intrinsic critical collapse in Kerr media. Here, we provide a possibility for supporting 2D matter-wave solitons and vortices in an extended setting — the cubic and quintic model — by introducing another nonlinear lattice whose period is controllable and can be different from its cubic counterpart, to its quintic nonlinearity, therefore making a fully “nonlinear quasi-crystal”.A variational approximation based on Gaussian ansatz is developed for the fundamental solitons and in particular, their stability exactly follows the inverted Vakhitov–Kolokolov stability criterion, whereas the vortex solitons are only studied by means of numerical methods. Stability regions for two types of localized mode — the fundamental and vortex solitons — are provided. A noteworthy feature of the localized solutions is that the vortex solitons are stable only when the period of the quintic nonlinear lattice is the same as the cubic one or when the quintic nonlinearity is constant, while the stable fundamental solitons can be created under looser conditions. Our physical setting (cubic-quintic model) is in the framework of the Gross–Pitaevskii equation or nonlinear Schrödinger equation, the predicted localized modes thus may be implemented in Bose–Einstein condensates and nonlinear optical media with tunable cubic and quintic nonlinearities.     

Dark Solitons for the Defocusing Cubic Nonlinear Schrödinger Equation with the Spatially Periodic Potential and Nonlinearity

We study the existence of dark solitons of the defocusing cubic nonlinear Schrödinger (NLS) eqaution with the spatially-periodic potential and nonlinearity. Firstly, we propose six families of upper and lower solutions of the dynamical systems arising from the stationary defocusing NLS equation. Secondly, by regarding a dark soliton as a heteroclinic orbit of the Poincaré map, we present some constraint conditions for the periodic potential and nonlinearity to show the existence of stationary dark solitons of the defocusing NLS equation for six different cases in terms of the theory of strict lower and upper solutions and the dynamics of planar homeomorphisms. Finally, we give the explicit dark solitons of the defocusing NLS equation with the chosen periodic potential and nonlinearity.     

Solitons in PT-symmetric potential with competing nonlinearity

We investigate the effect of competing nonlinearities on beam dynamics in PT-symmetric potentials. In particular, we consider the stationary nonlinear Schrödinger equation (NLSE) in one dimension with competing cubic and generalized nonlinearity in the presence of a PT-symmetric potential. Closed form solutions for localized states are obtained. These solitons are shown to be stable over a wide range of potential parameters. The transverse power flow associated with these complex solitons is also examined.     

Abundant different types of exact soliton solution to the (4+1)-dimensional Fokas and (2+1)-dimensional breaking soliton equations

The prime objective of this paper is to explore the new exact soliton solutions to the higher-dimensional nonlinear Fokas equation and(2+1)-dimensional breaking soliton equations via a generalized exponential rational function(GERF) method. Many different kinds of exact soliton solution are obtained, all of which are completely novel and have never been reported in the literature before. The dynamical behaviors of some obtained exact soliton solutions are also demonstrated by a choice of appropriate values of the free constants that aid in understanding the nonlinear complex phenomena of such equations. These exact soliton solutions are observed in the shapes of different dynamical structures of localized solitary wave solutions, singular-form solitons, single solitons,double solitons, triple solitons, bell-shaped solitons, combo singular solitons, breather-type solitons,elastic interactions between triple solitons and kink waves, and elastic interactions between diverse solitons and kink waves. Because of the reduction in symbolic computation work and the additional constructed closed-form solutions, it is observed that the suggested technique is effective, robust, and straightforward. Moreover, several other types of higher-dimensional nonlinear evolution equation can be solved using the powerful GERF technique.     

Localized nonlinear waves in combined time-dependent magnetic–optical potentials with spatiotemporally modulated nonlinearities

We construct explicit novel solutions of the nonlinear Schrödinger equation with spatiotemporal modulation of the nonlinearities and potentials. By using a modified similarity transformation we explore some localized nonlinearities and combined time-dependent magnetic–optical potentials in the form of linear-lattice ones and harmonic-lattice ones. Several families of exact localized nonlinear wave solutions in terms of Mathieu and elliptic functions corresponding to these potentials are then studied, such as snakelike solitons and breathing solitons. The stability of the obtained localized nonlinear wave solutions is investigated numerically such that some stable solutions are found.     

Exact solutions for bright and dark solitons in spatially inhomogeneous nonlinearity

We present exact analytical results for bright and dark solitons in a type of one-dimensional spatially inhomogeneous nonlinearity. We show that the competition between a homogeneous self-defocusing (SDF) nonlinearity and a localized self-focusing (SF) nonlinearity supports stable fundamental bright solitons. For a specific choice of the nonlinear parameters, exact analytical solutions for fundamental bright solitons have been obtained. By applying both variational approximation and Vakhitov-Kolokolov stability criterion, it is found that exact fundamental bright solitons are stable. Our analytical results are also confirmed numerically. Additionally, we show that a homogeneous SF nonlinearity modulated by a localized SF nonlinearity allows the existence of exact dark solitons, for certain special cases of nonlinear parameters. By making use of linear stability analysis and direct numerical simulation, it is found that these exact dark solitons are linearly unstable.     

Novel localized wave solutions of the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation

Based on a special transformation that we introduce, the N-soliton solution of the (2+1)-dimensional Boiti-Leon-Manna-Pempinelli equation is constructed. By applying the long wave limit and restricting certain conjugation conditions to the related solitons, some novel localized wave solutions are obtained, which contain higher-order breathers and lumps as well as their interactions. In particular, by choosing appropriate parameters involved in the N-solitons, two interaction solutions mixed by a bell-shaped soliton and one breather or by a bell-shaped soliton and one lump are constructed from the 3-soliton solution. Five solutions including two breathers, two lumps, and interaction solutions between one breather and two bell-shaped solitons, one breather and one lump, or one lump and two bell-shaped solitons are constructed from the 4-soliton solution. Five interaction solutions mixed by one breather/lump and three bell-shaped solitons, two breathers/lumps and a bell-shaped soliton, as well as mixing with one lump, one breather and a bell-shaped soliton are constructed from the 5-soliton solution. To study the behaviors that the obtained interaction solutions may have, we present some illustrative numerical simulations, which demonstrate that the choice of the parameters has a great impacts on the types of the solutions and their propagation properties. The method proposed can be effectively used to construct localized interaction solutions of many nonlinear evolution equations. The results obtained may help related experts to understand and study the interaction phenomena of nonlinear localized waves during propagations.     

Bilinear Forms and Dark-Dark Solitons for the Coupled Cubic-Quintic Nonlinear SchrÕdinger Equations with Variable Coefficients in a Twin-Core Optical Fiber or Non-Kerr Medium*

Twin-core optical fibers are applied in such fields as the optical sensing and optical communication,and propagation of the pulses,Gauss beams and laser beams in the non-Kerr media is reported.Studied in this paper are the coupled cubic-quintic nonlinear Schrodinger equations with variable coefficients,which describe the effects of quintic nonlinearity for the ultrashort optical pulse propagation in a twin-core optical fiber or non-Kerr medium.Based on the integrable conditions,bilinear forms are derived,and dark-dark soliton solutions can be constructed in terms of the Gramian via the Kadomtsev-Petviashvili hierarchy reduction.Propagation and interaction of the dark-dark solitons are presented and discussed through the graphic analysis.With different values of the delayed nonlinear response effect b(z),where z represents direction of the propagation,the linear-and parabolic-shaped one dark-dark soltions can be derived.Interactions between the parabolic-and periodic-shaped two dark-dark solitons are presented with b(z) as the linear and periodic functions,respectively.Directions of velocities of the two dark-dark solitons vary with z and the amplitudes of the solitons remain unchanged can be observed.Interactions between the two dark-dark solitons of different types are displayed,and we observe that the velocity of one soliton is zero and direction of the velocity of the other soliton vary with z.We find that those interactions are elastic.     

On nonlinear effects in magnetic chains

The effect of phonon unharmonism and nonlinearity in exchange integrals on soliton excitations in ferromagnetic chains in the classical and long-wave limit are studied. It has been first shown that the unharmonic effect leads to a system of coupled Boussinesque and nonlinear Schrödinger equations allowing two types of soliton solutions. The nonlinear effect on the other hand results nonlinear Schrödinger equation with saturable nonlinearity admitting stable solitons in higher dimensional models.     

Oscillating solitons in nonlinear optics

Oscillating solitons are obtained in nonlinear optics. Analytical study of the variable-coefficient nonlinear Schrödinger equation, which is used to describe the soliton propagation in those systems, is carried out using the Hirota’s bilinear method. The bilinear forms and analytic soliton solutions are derived, and the relevant properties and features of oscillating solitons are illustrated. Oscillating solitons are controlled by the reciprocal of the group velocity and Kerr nonlinearity. Results of this paper will be valuable to the study of dispersion-managed optical communication system and mode-locked fibre lasers.